If the sole means of accessing an online salon software system is via a web browser, and there's no "standalone" application running on your computer with a local copy of all your data, then you would pretty much be out of luck if the Internet is down.
You're also relying on third party servers for storing your appointment details and client data, so would need to make sure that the company providing the service were registered with the Information Commissioner's Office under the Data Protection Act, and had suitable security procedures in place to make sure that only you had access to your client database (e.g. making sure that all access to sensitive information is over a secure connection). If going the "online" route, it would also be wise to go with a company that runs their application on at least two servers on separate sites, in case there was ever an issue with either server or data centre (modern enterprise Web applications can be "clustered" to run on more than one server).
Personally, I think the best of both worlds would be a hybrid of both a standalone system and an online system; the main computer at the salon running a standalone application which maintains a local copy of all appointment details and client details, but constantly synchronises with the online application while the Internet connection is up and running.
Then, if your broadband goes down, you've still got access to your appointment details and client database. But you'd still have all the advantages of an online system too, e.g. being able to securely access the online version of the software from any computer connected to the internet anywhere in the world, and being able to accept online bookings via the web.
Phorest is great to use, sorry if this was already mentioned but I didnt have time to read it all. Its like 'salon systems for dummies'. You can do online booking and every client gets a loyalty tag where they build up points. Very easy to buy gift vouchers and pre pay treatments etc
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